Who has the advantage in the 1-11 game to reach 56?

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    2016
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SUMMARY

The game described involves two players alternately selecting integers to reach a target of 56. The first player has a strategic advantage by starting the game, allowing them to control the pace and direction of play. Players can choose any integer from 1 to 11, and the key to winning lies in forcing the opponent into a position where they cannot win. The optimal strategy involves calculating the winning positions, specifically targeting numbers that leave the opponent with limited options.

PREREQUISITES
  • Understanding of basic game theory concepts
  • Familiarity with strategic decision-making
  • Knowledge of integer properties and arithmetic
  • Ability to analyze winning and losing positions in games
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  • Study game theory strategies for two-player games
  • Learn about combinatorial game theory and its applications
  • Explore mathematical induction techniques to analyze game outcomes
  • Investigate similar games and their optimal strategies, such as Nim or Chomp
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Mathematicians, game theorists, educators, and anyone interested in strategic gameplay and mathematical problem-solving.

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Here is this week's POTW:

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Two players play the following game. The first player selects any integer from 1 to 11 inclusive. The second player adds any positive integer from 1 to 11 inclusive to the number selected by the first player. They continue in this manner alternately. The player who reaches 56 wins the game. Which player has the advantage?

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Remember to read the http://www.mathhelpboards.com/showthread.php?772-Problem-of-the-Week-%28POTW%29-Procedure-and-Guidelines to find out how to http://www.mathhelpboards.com/forms.php?do=form&fid=2!
 
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Congratulations to Kiwi, kaliprasad, and Fallen Angel for their correct solutions! Kiwi's solution follows:

Player 1 always wins by following this sequence:

Player 1 plays 8
Player 2 plays 1-11 bringing the total to 9-19
Player 1 brings the total to 20
Player 2 plays 1-11 bringing the total to 21-31
Player 1 brings the total to 32
Player 2 plays 1-11 bringing the total to 33-43
Player 1 brings the total to 44
Player 2 plays 1-11 bringing the total to 45-55
Player 1 brings the total to 56 and wins
 

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